Step 1: Write the given.
We are told $\sin\theta+\cos\theta=1$ and we must find $\sin\theta\cos\theta$.
Step 2: Square both sides.
Squaring removes the sum and brings in a product term: $(\sin\theta+\cos\theta)^2=1^2$.
Step 3: Expand the left side.
$(\sin\theta+\cos\theta)^2=\sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta$.
Step 4: Use the Pythagorean identity.
Since $\sin^2\theta+\cos^2\theta=1$, the left side becomes $1+2\sin\theta\cos\theta$.
Step 5: Set equal to the right side.
So $1+2\sin\theta\cos\theta=1$.
Step 6: Solve for the product.
Subtract $1$ from both sides: $2\sin\theta\cos\theta=0$, so $\sin\theta\cos\theta=0$.
Step 7: State the answer.
\[ \boxed{0} \]